Let $\Delta$ be a set of infinite subsets of reals, suppose for any elements $X\neq Y$ of $\Delta$, $X\cap Y$ is finite. Does that imply anything about the cardinality of $\Delta$? Notice that if we are looking at a set of infinite subsets of naturals, then the set can be as small as $\omega$ and as large as $2^\omega$.
I am interested in knowing an answer assuming the axiom of choice, because I imagine without the axiom of choice things are pretty wild. For instance if $\mathbb{R}$ is a countable union of pairwise disjoint countable sets, then we could have $\Delta$ being countable. Or if we partition $\mathbb{R}$ into more equivalence classes than $|\mathbb{R}|$ itself, then we could have $\Delta$ being "larger" than $|\mathbb{R}|$.
We get the same range of possibilities as for almost disjoint families in $\wp(\omega)$.
Let $\mathscr{D}$ be an almost disjoint family of infinite subsets of $\Bbb R$. For each $D\in\mathscr{D}$ choose a countably infinite $C_D\subseteq\Bbb R$ such that $C_D\subseteq D$, and let $\mathscr{C}=\{C_D:D\in\mathscr{D}\}$; clearly $\mathscr{C}$ is almost disjoint. Moreover, if $D,E\in\mathscr{D}$, and $D\ne E$, then $D\cap E$ is finite, so $C_D\cap C_E$ is finite, and $C_D\ne C_E$. Thus, $|\mathscr{D}|=|\mathscr{C}|\le 2^\omega$.